[At the moderators' suggestion, I have resubmitted an earlier answer with more detail on my sources]

Our knowledge of disease in the Roman world is limited by the nature of our literary sources (above all, the works of Galen), which often describe symptoms in a vague or stylized manner that makes identification difficult.

That said, if I were traveling to ancient Rome, I would have myself vaccinated for:

(a) Malaria. The single most persistent and deadly threat around Rome, Malaria seems to have been especially prevalent in the imperial era, when it became hyperendemic in the populace. Galen refers to the disease by its symptoms (quotidian fever and semitertian fever) and notes that children were especially prone to quotidian fever (a fever that spikes daily) - an observation that coheres nicely with our knowledge of malaria. According to Scheidel, children and recent immigrants (including, presumably, time travelers) were especially susceptible, since most Roman adults would have developed immunity to the local strains.

(b) Typhoid Fever. Though harder to pin down in Galen, Typhoid was (according to the entry on "Disease" in Brill's New Pauly) probably also prevalent in ancient Rome, and very nasty - the infamous Plague of Athens is sometimes identified as a strain of Typhoid.

(c) Tuberculosis. Tuberculosis (described by Galen as phthisis, an incurable wasting disease) was common in ancient Rome; in one of his epigrams (1.10), for example, the Roman poet Martial relates how an unscrupulous young man began to court a rich woman with a tubercular cough.

(d) Polio. The evidence is not wholly clear, but Galen describes a disease known for causing paralysis that seems (or so the author of the Pauly article thinks) to be polio.

(e) Smallpox. The vaccine isn't easy to get these days; but if I were traveling to Rome in the late second century CE or after, I'd be sure to have myself inoculated, since Smallpox may (according to Scheidel) have become endemic in Rome after the Antonine Plague.

According to the Oxford Classical Dictionary entry on "Disease", chickenpox, mumps, and whooping cough were also present in ancient Rome, so I'd have to make sure my vaccinations for those were up to date. If I were planning a particularly adventurous visit, I might also look into the experimental vaccines for chlamydia and herpes, the only STDs known to be prevalent in antiquity.

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Sources

Ian Johnson (trans.), Galen on Diseases and Symptoms (Cambridge, 2006)

A. S. Pease, "Some remarks on the diagnosis and treatment of tuberculosis in antiquity" Isis 31 (1940): 380–93.

Walter Scheidel, "Germs for Rome" in C. Edwards and G. Woolf, eds., Rome the Cosmopolis (Cambridge, 2003), 158-76

Back when I did primary care, I used to do travel medicine, which essentially was looking up CDC alerts and providing the appropriate vaccinations, as well as providing gastroenteritis treatment and malaria prophylaxis. With that experience, I'd like to offer my "Time Travelers Clinic" for Imperial Rome:

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1. Mandatory vaccinations -- TDap, MMR, Varicella, Polio, influenza (quadrivalent), Hep A, Typhoid, Hep B, Polio, Meningococcus, Rabies, Yellow fever, smallpox (actually is still produced to maintain stocks; still made from cowpox)

2. Optional vaccines -- BCG (TB vaccine, helpful in children in endemic areas, unclear benefit in an adult), plague (imperfect protection, and it's live attenuated so chance of spreading)

3. Malaria prophylaxis -- not likely to have any chloroquine resistance, so barring contraindictions would use this. Otherwise would prescribe Malarone

4. Malaria treatment -- every traveler would be prescribed an emergency course of ACT (artemesin combo therapy) in case they come down with symptoms of malaria

5. Gastroenteritis treatment -- the biggest killer is likely to be a lowly gastroenteritis (likely bacillary). Every patient would get a course of ciprofloxacin to take in case of symptoms of gastroenteritis. Will also have oral rehydration salts.

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Your consultation in the Time Travelers Clinic will be 200 dollars, likely covered by insurance. I should note that many of these vaccines are NOT covered by insurance (such as yellow fever, smallpox, and the option vaccines); you will need to pay out of pocket.

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Save travels!